Tight Staircase Bounds for Cyclic Subsets below Dirac's Threshold

Abstract: Let $\operatorname{Cyc}(G)$ denote the number of cyclic subsets in a graph $G$, which are subsets that induce a Hamiltonian subgraph. Draganić, Keevash and Müyesser recently proved that every regular Dirac graph has $Ω(2<sup>n)$ cyclic subsets, resolving a problem of Erdős and Faudree. We determine the sharp asymptotic lower bound throughout the linear range below Dirac's threshold. Let $G$ be an $n$-vertex $d$-regular graph with $d=Ω(n)$ and $d&lt;n/2$, then $$\operatorname{Cyc}(G)\ge (q-o(1))2^{n/q}, \quad \text{where } \quad q=\left\lfloor \frac{n}{d+1}\right\rfloor \ge 2.$$ This bound is asymptotically best possible, including the leading coefficient $q$, as witnessed at the staircase levels by the disjoint union of $q$ equal cliques. Consequently, the optimal exponential rate changes by discrete jumps as $d$ crosses the thresholds $n/k$, rather than varying smoothly with $d$. We also prove the optimal exponential rate at the Dirac boundary: every $n$-vertex $n/2$-regular graph satisfies $\operatorname{Cyc}(G)\ge 2<sup>{(1-o(1))n},$ which is sharp up to a subexponential factor by $K_{n/2,n/2}$.

• The paper determines the exact asymptotic lower bound for cyclic subsets in d-regular graphs, showing a discrete staircase behavior based on the structure of disjoint cliques.
• It employs Szemerédi's Regularity Lemma and reduced graph analysis to precisely enumerate Hamiltonian induced subgraphs, revealing exponential rates that jump at certain degree thresholds.
• The results not only refine classical Hamiltonicity results but also have significant implications for extremal combinatorics, informing future research on graph robustness and substructure enumeration.

Tight Staircase Bounds for Cyclic Subsets in Regular Graphs Below Dirac's Threshold

Introduction and Context

The enumeration of Hamiltonian cycles and, more generally, Hamiltonian induced subgraphs in dense graphs has been a central topic in extremal and enumerative combinatorics. Building on Dirac's theorem, which states that any $n$-vertex graph with minimum degree at least $n/2$ is Hamiltonian, recent developments have focused on quantified versions and robustness: not merely the existence, but the guaranteed multiplicity of Hamiltonian substructures. The focus of this work is on cyclic subsets, i.e., vertex subsets whose induced subgraph contains a Hamiltonian cycle. The main research question is to determine sharp lower bounds on the number of cyclic subsets, denoted $\operatorname{Cyc}(G)$, in $n$-vertex $d$-regular graphs with $d = \Omega(n)$, specifically throughout the linear range $d < n/2$—that is, below Dirac's threshold.

Earlier work by Draganić, Keevash, and Müyesser established that every regular Dirac graph admits exponentially many cyclic subsets—$\Omega(2^n)$—thus affirmatively resolving a conjecture of Erdős and Faudree. The present paper by Liu, Niu, Wang, and Yan determines, for the first time, the exact asymptotic lower bound for $\operatorname{Cyc}(G)$ in the linear regime $d < n/2$, revealing a staircase behavior as a function of $n/2$0 rather than a smooth transition.

Main Results

The core results can be summarized as follows:

• Sharp Lower Bound for Cyclic Subsets: For an $n/2$1-vertex $n/2$2-regular graph with $n/2$3 and $n/2$4, the number of cyclic subsets satisfies

$n/2$5

where $n/2$6 is the maximal number of disjoint components that a $n/2$7-regular graph can have. This result is tight, including the leading coefficient $n/2$8, as is demonstrated by the extremal construction given by the disjoint union of $n/2$9 cliques of size $\operatorname{Cyc}(G)$0 (2607.06551).

• Staircase Phenomenon: The exponential rate $\operatorname{Cyc}(G)$1 transitions in discrete jumps as $\operatorname{Cyc}(G)$2 crosses the thresholds $\operatorname{Cyc}(G)$3. Thus, unlike the monotonic increase of Hamiltonicity expected from increasing degree, the enumeration of cyclic subsets exhibits discontinuities—referred to as the staircase phenomenon.
• Boundary at Dirac's Threshold: For $\operatorname{Cyc}(G)$4, it is shown that

$\operatorname{Cyc}(G)$5

with this bound being sharp up to a subexponential factor. As an extremal example, the balanced complete bipartite graph $\operatorname{Cyc}(G)$6 achieves $\operatorname{Cyc}(G)$7.

Techniques and Proof Strategy

The methodology relies deeply on Szemerédi's Regularity Lemma, reduced graph analysis, and modern structural techniques of extremal graph theory. The proof proceeds by passing to a regular partition and examining the structure of the reduced graph:

1. Reduced Graph Trichotomy: For the reduced graph $\operatorname{Cyc}(G)$8 arising from the regularity partition, a trichotomy is established:
   • The existence of a large connected matching (enabling concatenation of paths into cycles in $\operatorname{Cyc}(G)$9);
   • $n$0 balanced components, corresponding to the extremal clique decomposition;
   • $n$1 near-equal components, which force exponential surplus over the baseline via compensating edges.
2. Enumeration in Each Case:
   • Connected Matching (Non-Degenerate Case): With a sufficiently large matching in $n$2, exponentially many cyclic subsets are constructed by combining Hamiltonian paths in regular pairs and linking them with short paths.
   • Balanced Dense Components (Staircase Extremal Case): When $n$3 consists of $n$4 components, almost every subset within each block is cyclic, leveraging Chvátal's theorem and blockwise expansion.
   • Near-Critical Components: When $n$5 has $n$6 components, matching arguments ensure sufficient inter-block connectivity to create exponentially more cyclic subsets than the staircase baseline.

The paper develops advanced probabilistic and structural lemmas to show that, across all cases, the claimed lower bound not only holds but is asymptotically tight.

Strong Numerical and Structural Claims

• Optimal Leading Coefficient: For every fixed value of $n$7, the leading coefficient $n$8 in $n$9 cannot be improved, as witnessed by the clique union example.
• Discontinuity at Thresholds: The exponential base $d$0 remains stable as $d$1 varies in $d$2 and jumps discretely when $d$3 crosses $d$4 barriers. This contradicts heuristics predicting smooth dependence on $d$5.
• Boundary Tightness: At $d$6, $d$7 is shown to be sharp.

Theoretical and Practical Implications

From a theoretical perspective, these results provide a comprehensive and exact answer to the enumerative extension of Dirac-type Hamiltonicity, demonstrating a new robustness phenomenon for vertex-inductive Hamiltonicity under strict regularity. The staircase pattern, depending on the number of possible disjoint components, reveals an unexpected discrete structure underlying the Hamiltonicity enumeration below Dirac's threshold.

This also informs related questions in extremal combinatorics, such as packing, resilience, and local-to-global principles in graph robustness. For practical applications, particularly in random or pseudorandom regular graphs, the results establish boundary rates for subgraph Hamiltonicity and give explicit constructions that match the lower bound.

The paper raises several natural open problems, including exact minimization for other ranges, size-sensitive bounds (counting cyclic subsets of given cardinality), and extension to more general degree constraints (e.g., minimum degree rather than regularity).

Conclusion

The paper establishes precise staircase lower bounds for the number of cyclic subsets in $d$8-regular $d$9-vertex graphs with $d = \Omega(n)$0, showing that the optimal exponential rate changes in discrete steps depending on the value of $d = \Omega(n)$1. This sharply resolves longstanding problems originating with Erdős and Faudree and demonstrates that both global regularity and the component structure at sub-Dirac degrees critically determine the enumeration of Hamiltonian subgraphs. The techniques and findings lay a foundation for new research avenues in extremal graph theory and combinatorial enumeration (2607.06551).